# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #( 3pi)/8 #, and the triangle's area is #16 #. What is the area of the triangle's incircle?

The area of the incircle is

The area of the triangle is

The angle

The angle

The angle

The sine rule is

So,

Let the height of the triangle be

The area of the triangle is

But,

So,

Therefore,

The radius of the incircle is

The area of the incircle is

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To find the area of the triangle's incircle, you can use the formula:

Area of incircle = semiperimeter of triangle × inradius

First, find the semiperimeter (s) of the triangle. Semiperimeter is half the sum of the triangle's three sides.

s = (AB + BC + AC) / 2

Next, find the lengths of the sides using the given angles and the area of the triangle. You can use trigonometry to find the side lengths.

Once you have the side lengths, you can use Heron's formula to find the area of the triangle:

Area of triangle = √(s(s - AB)(s - BC)(s - AC))

Given that the area of the triangle is 16, you can set up the equation:

16 = √(s(s - AB)(s - BC)(s - AC))

Now, solve for s.

After finding the semiperimeter, you can use it to find the inradius. The formula for the inradius (r) of a triangle in terms of its sides (a, b, and c) and its semiperimeter (s) is:

inradius (r) = Area of triangle / semiperimeter

Once you find the inradius, you can then calculate the area of the incircle using the formula:

Area of incircle = πr^2

Substitute the value of the inradius into the formula to find the area of the incircle.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A circle has a center at #(7 ,9 )# and passes through #(1 ,1 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?
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